Revisions to Certain asymptotics involving double infinite sum

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edited Aug 14, 2013 at 22:45

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Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, how does $S(n)$ behave asymptotically, e.g., if it is asymptotically a power function? If yes, what is the exponent?

Remarks:

When $\alpha=\beta$, this problem can be resolved using an integral approximation argument (rewriting the sum as a double integral by replacing $\frac{i}{n}$ with $\frac{[nx]+1}{n}$, $\frac{j}{n}$ with $\frac{[ny]+1}{n}$ and letting $n\rightarrow\infty$ through the Dominated Convergence Theorem) which yields $S(n)\sim c n^{2-2\alpha}$ for some $c>0$. But when $\alpha<\beta$, the similar argument seems difficult to apply due to the non-homogeneity of the function $g(x,y)=(x^{\alpha} +y^{\beta})^{-1}$.

It seems that if we do have $S(n)\sim cn^{2-2\gamma}$ for some $\gamma$, then $\alpha\le \gamma\le \beta$. Furthermore, by Jensen's inequality, we have for any $0<\rho<1$, $i^\alpha+j^\beta\ge c i^{-\alpha\rho}j^{-\beta(1-\rho)}$$i^\alpha+j^\beta\ge c i^{\alpha\rho}j^{\beta(1-\rho)}$ (now $g(x,y)= x^{-\rho\alpha}y^{-(1-\rho)\beta}$ is homogeneous, and an integral approximation argument applies provided $\alpha\rho\in (1/2,3/4)$, $\beta(1-\rho)\in (1/2,3/4)$), we should have $ \gamma\ge\rho\alpha+(1-\rho)\beta. $ By taking $\rho$ close to $1/(2\alpha)$, we expect that $\gamma\ge \beta+(\alpha-\beta)/(2\alpha)$.

Update: Matt shows below that $cn^{2-2\gamma}\le S(n)\le C n^{2-2\gamma}$, where $$\gamma=\beta+\frac{\alpha-\beta}{2\alpha}=\rho\alpha+(1-\rho)\beta\in (\alpha,\beta),$$ with $\rho=\frac{1}{2\alpha}$. Now the problem becomes whether one can show that $S(n)\sim cn^{2-2\gamma}$ where $\gamma$ is given as above.

Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, how does $S(n)$ behave asymptotically, e.g., if it is asymptotically a power function? If yes, what is the exponent?

Remarks:

When $\alpha=\beta$, this problem can be resolved using an integral approximation argument (rewriting the sum as a double integral by replacing $\frac{i}{n}$ with $\frac{[nx]+1}{n}$, $\frac{j}{n}$ with $\frac{[ny]+1}{n}$ and letting $n\rightarrow\infty$ through the Dominated Convergence Theorem) which yields $S(n)\sim c n^{2-2\alpha}$ for some $c>0$. But when $\alpha<\beta$, the similar argument seems difficult to apply due to the non-homogeneity of the function $g(x,y)=(x^{\alpha} +y^{\beta})^{-1}$.

It seems that if we do have $S(n)\sim cn^{2-2\gamma}$ for some $\gamma$, then $\alpha\le \gamma\le \beta$. Furthermore, by Jensen's inequality, we have for any $0<\rho<1$, $i^\alpha+j^\beta\ge c i^{-\alpha\rho}j^{-\beta(1-\rho)}$ (now $g(x,y)= x^{-\rho\alpha}y^{-(1-\rho)\beta}$ is homogeneous, and an integral approximation argument applies provided $\alpha\rho\in (1/2,3/4)$, $\beta(1-\rho)\in (1/2,3/4)$), we should have $ \gamma\ge\rho\alpha+(1-\rho)\beta. $ By taking $\rho$ close to $1/(2\alpha)$, we expect that $\gamma\ge \beta+(\alpha-\beta)/(2\alpha)$.

Update: Matt shows below that $cn^{2-2\gamma}\le S(n)\le C n^{2-2\gamma}$, where $$\gamma=\beta+\frac{\alpha-\beta}{2\alpha}=\rho\alpha+(1-\rho)\beta\in (\alpha,\beta),$$ with $\rho=\frac{1}{2\alpha}$. Now the problem becomes whether one can show that $S(n)\sim cn^{2-2\gamma}$ where $\gamma$ is given as above.

Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, how does $S(n)$ behave asymptotically, e.g., if it is asymptotically a power function? If yes, what is the exponent?

Remarks:

When $\alpha=\beta$, this problem can be resolved using an integral approximation argument (rewriting the sum as a double integral by replacing $\frac{i}{n}$ with $\frac{[nx]+1}{n}$, $\frac{j}{n}$ with $\frac{[ny]+1}{n}$ and letting $n\rightarrow\infty$ through the Dominated Convergence Theorem) which yields $S(n)\sim c n^{2-2\alpha}$ for some $c>0$. But when $\alpha<\beta$, the similar argument seems difficult to apply due to the non-homogeneity of the function $g(x,y)=(x^{\alpha} +y^{\beta})^{-1}$.

It seems that if we do have $S(n)\sim cn^{2-2\gamma}$ for some $\gamma$, then $\alpha\le \gamma\le \beta$. Furthermore, by Jensen's inequality, we have for any $0<\rho<1$, $i^\alpha+j^\beta\ge c i^{\alpha\rho}j^{\beta(1-\rho)}$ (now $g(x,y)= x^{-\rho\alpha}y^{-(1-\rho)\beta}$ is homogeneous, and an integral approximation argument applies provided $\alpha\rho\in (1/2,3/4)$, $\beta(1-\rho)\in (1/2,3/4)$), we should have $ \gamma\ge\rho\alpha+(1-\rho)\beta. $ By taking $\rho$ close to $1/(2\alpha)$, we expect that $\gamma\ge \beta+(\alpha-\beta)/(2\alpha)$.

Update: Matt shows below that $cn^{2-2\gamma}\le S(n)\le C n^{2-2\gamma}$, where $$\gamma=\beta+\frac{\alpha-\beta}{2\alpha}=\rho\alpha+(1-\rho)\beta\in (\alpha,\beta),$$ with $\rho=\frac{1}{2\alpha}$. Now the problem becomes whether one can show that $S(n)\sim cn^{2-2\gamma}$ where $\gamma$ is given as above.

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edited Aug 2, 2013 at 12:58

Uchiha

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Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, how does $S(n)$ behave asymptotically, e.g., if it is asymptotically a power function? If yes, what is the exponent?

Remarks:

When $\alpha=\beta$, this problem can be resolved using an integral approximation argument (rewriting the sum as a double integral by replacing $\frac{i}{n}$ with $\frac{[nx]+1}{n}$, $\frac{j}{n}$ with $\frac{[ny]+1}{n}$ and letting $n\rightarrow\infty$ through the Dominated Convergence Theorem) which yields $S(n)\sim c n^{2-2\alpha}$ for some $c>0$. But when $\alpha<\beta$, the similar argument seems difficult to apply due to the non-homogeneity of the function $g(x,y)=(x^{\alpha} +y^{\beta})^{-1}$.

It seems that if we do have $S(n)\sim cn^{2-2\gamma}$ for some $\gamma$, then $\alpha\le \gamma\le \beta$. Furthermore, by Jensen's inequality, we have for any $0<\rho<1$, $i^\alpha+j^\beta\ge c i^{-\alpha\rho}j^{-\beta(1-\rho)}$ (now $g(x,y)= x^{-\rho\alpha}y^{-(1-\rho)\beta}$ is homogeneous, and an integral approximation argument applies provided $\alpha\rho\in (1/2,3/4)$, $\beta(1-\rho)\in (1/2,3/4)$), we should have $ \gamma\ge\rho\alpha+(1-\rho)\beta. $ By taking $\rho$ close to $1/(2\alpha)$, we expect that $\gamma\ge \beta+(\alpha-\beta)/(2\alpha)$.

Update: Matt shows below that $cn^{2-2\gamma}\le S(n)\le C n^{2-2\gamma}$, where $$\gamma=\beta+\frac{\alpha-\beta}{2\alpha}=\rho\alpha+(1-\rho)\beta\in (\alpha,\beta),$$ with $\rho=\frac{1}{2\alpha}$. Now the problem becomes whether one can show that $S(n)\sim cn^{2-2\gamma}$ where $\gamma$ is given as above.

Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, how does $S(n)$ behave asymptotically, e.g., if it is asymptotically a power function? If yes, what is the exponent?

Remarks:

When $\alpha=\beta$, this problem can be resolved using an integral approximation argument (rewriting the sum as a double integral by replacing $\frac{i}{n}$ with $\frac{[nx]+1}{n}$, $\frac{j}{n}$ with $\frac{[ny]+1}{n}$ and letting $n\rightarrow\infty$ through the Dominated Convergence Theorem) which yields $S(n)\sim c n^{2-2\alpha}$ for some $c>0$. But when $\alpha<\beta$, the similar argument seems difficult to apply due to the non-homogeneity of the function $g(x,y)=(x^{\alpha} +y^{\beta})^{-1}$.

It seems that if we do have $S(n)\sim cn^{2-2\gamma}$ for some $\gamma$, then $\alpha\le \gamma\le \beta$. Furthermore, by Jensen's inequality, we have for any $0<\rho<1$, $i^\alpha+j^\beta\ge c i^{-\alpha\rho}j^{-\beta(1-\rho)}$ (now $g(x,y)= x^{-\rho\alpha}y^{-(1-\rho)\beta}$ is homogeneous, and an integral approximation argument applies provided $\alpha\rho\in (1/2,3/4)$, $\beta(1-\rho)\in (1/2,3/4)$), we should have $ \gamma\ge\rho\alpha+(1-\rho)\beta. $ By taking $\rho$ close to $1/(2\alpha)$, we expect that $\gamma\ge \beta+(\alpha-\beta)/(2\alpha)$.

Update: Matt shows below that $cn^{2-2\gamma}\le S(n)\le C n^{2-2\gamma}$, where $$\gamma=\beta+\frac{\alpha-\beta}{2\alpha}=\rho\alpha+(1-\rho)\beta\in (\alpha,\beta),$$ with $\rho=\frac{1}{2\alpha}$.

Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, how does $S(n)$ behave asymptotically, e.g., if it is asymptotically a power function? If yes, what is the exponent?

Remarks:

When $\alpha=\beta$, this problem can be resolved using an integral approximation argument (rewriting the sum as a double integral by replacing $\frac{i}{n}$ with $\frac{[nx]+1}{n}$, $\frac{j}{n}$ with $\frac{[ny]+1}{n}$ and letting $n\rightarrow\infty$ through the Dominated Convergence Theorem) which yields $S(n)\sim c n^{2-2\alpha}$ for some $c>0$. But when $\alpha<\beta$, the similar argument seems difficult to apply due to the non-homogeneity of the function $g(x,y)=(x^{\alpha} +y^{\beta})^{-1}$.

It seems that if we do have $S(n)\sim cn^{2-2\gamma}$ for some $\gamma$, then $\alpha\le \gamma\le \beta$. Furthermore, by Jensen's inequality, we have for any $0<\rho<1$, $i^\alpha+j^\beta\ge c i^{-\alpha\rho}j^{-\beta(1-\rho)}$ (now $g(x,y)= x^{-\rho\alpha}y^{-(1-\rho)\beta}$ is homogeneous, and an integral approximation argument applies provided $\alpha\rho\in (1/2,3/4)$, $\beta(1-\rho)\in (1/2,3/4)$), we should have $ \gamma\ge\rho\alpha+(1-\rho)\beta. $ By taking $\rho$ close to $1/(2\alpha)$, we expect that $\gamma\ge \beta+(\alpha-\beta)/(2\alpha)$.

Update: Matt shows below that $cn^{2-2\gamma}\le S(n)\le C n^{2-2\gamma}$, where $$\gamma=\beta+\frac{\alpha-\beta}{2\alpha}=\rho\alpha+(1-\rho)\beta\in (\alpha,\beta),$$ with $\rho=\frac{1}{2\alpha}$. Now the problem becomes whether one can show that $S(n)\sim cn^{2-2\gamma}$ where $\gamma$ is given as above.

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edited Aug 1, 2013 at 22:20

Uchiha

87
1
12

Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, how does $S(n)$ behave asymptotically, e.g., if it is asymptotically a power function? If yes, what is the exponent?

Remarks:

When $\alpha=\beta$, this problem can be resolved using an integral approximation argument (rewriting the sum as a double integral by replacing $\frac{i}{n}$ with $\frac{[nx]+1}{n}$, $\frac{j}{n}$ with $\frac{[ny]+1}{n}$ and letting $n\rightarrow\infty$ through the Dominated Convergence Theorem) which yields $S(n)\sim c n^{2-2\alpha}$ for some $c>0$. But when $\alpha<\beta$, the similar argument seems difficult to apply due to the non-homogeneity of the function $g(x,y)=(x^{\alpha} +y^{\beta})^{-1}$.

It seems that if we do have $S(n)\sim cn^{2-2\gamma}$ for some $\gamma$, then $\alpha\le \gamma\le \beta$. Furthermore, by Jensen's inequality, we have for any $\rho_1,\rho_2> 0$ and $\rho_1+\rho_2=1$ $0<\rho<1$, we have $i^\alpha+j^\beta\ge c i^{-\alpha\rho_1}j^{-\beta\rho_2}$ $i^\alpha+j^\beta\ge c i^{-\alpha\rho}j^{-\beta(1-\rho)}$ (now $g(x,y)= x^{-\rho_1\alpha}y^{-\rho_2\beta}$$g(x,y)= x^{-\rho\alpha}y^{-(1-\rho)\beta}$ is homogeneous of degree $-(\rho_1\alpha+\rho_2\beta)$, and an integral approximation argument applies provided $\alpha\rho_1\in (1/2,3/4)$$\alpha\rho\in (1/2,3/4)$, $\beta\rho_2\in (1/2,3/4)$$\beta(1-\rho)\in (1/2,3/4)$), we should have $ \gamma\ge\rho\alpha+(1-\rho)\beta. $ By taking $\gamma<\rho_1\alpha+\rho_2\beta$$\rho$ close to $1/(2\alpha)$, we expect that $\gamma\ge \beta+(\alpha-\beta)/(2\alpha)$.

Update: Matt shows below that $cn^{2-2\gamma}\le S(n)\le C n^{2-2\gamma}$, where $$\gamma=\beta+\frac{\alpha-\beta}{2\alpha}=\rho\alpha+(1-\rho)\beta\in (\alpha,\beta),$$ with $\rho=\frac{1}{2\alpha}$.

Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, how does $S(n)$ behave asymptotically, e.g., if it is asymptotically a power function? If yes, what is the exponent?

Remarks:

When $\alpha=\beta$, this problem can be resolved using an integral approximation argument (rewriting the sum as a double integral by replacing $\frac{i}{n}$ with $\frac{[nx]+1}{n}$, $\frac{j}{n}$ with $\frac{[ny]+1}{n}$ and letting $n\rightarrow\infty$ through the Dominated Convergence Theorem) which yields $S(n)\sim c n^{2-2\alpha}$ for some $c>0$. But when $\alpha<\beta$, the similar argument seems difficult to apply due to the non-homogeneity of the function $g(x,y)=(x^{\alpha} +y^{\beta})^{-1}$.

It seems that if we do have $S(n)\sim cn^{2-2\gamma}$ for some $\gamma$, then $\alpha\le \gamma\le \beta$. Furthermore, by Jensen's inequality, we have for any $\rho_1,\rho_2> 0$ and $\rho_1+\rho_2=1$, we have $i^\alpha+j^\beta\ge c i^{-\alpha\rho_1}j^{-\beta\rho_2}$ (now $g(x,y)= x^{-\rho_1\alpha}y^{-\rho_2\beta}$ is homogeneous of degree $-(\rho_1\alpha+\rho_2\beta)$, and an integral approximation argument applies provided $\alpha\rho_1\in (1/2,3/4)$, $\beta\rho_2\in (1/2,3/4)$), we should have $\gamma<\rho_1\alpha+\rho_2\beta$.

Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, how does $S(n)$ behave asymptotically, e.g., if it is asymptotically a power function? If yes, what is the exponent?

Remarks:

When $\alpha=\beta$, this problem can be resolved using an integral approximation argument (rewriting the sum as a double integral by replacing $\frac{i}{n}$ with $\frac{[nx]+1}{n}$, $\frac{j}{n}$ with $\frac{[ny]+1}{n}$ and letting $n\rightarrow\infty$ through the Dominated Convergence Theorem) which yields $S(n)\sim c n^{2-2\alpha}$ for some $c>0$. But when $\alpha<\beta$, the similar argument seems difficult to apply due to the non-homogeneity of the function $g(x,y)=(x^{\alpha} +y^{\beta})^{-1}$.

It seems that if we do have $S(n)\sim cn^{2-2\gamma}$ for some $\gamma$, then $\alpha\le \gamma\le \beta$. Furthermore, by Jensen's inequality, we have for any $0<\rho<1$, $i^\alpha+j^\beta\ge c i^{-\alpha\rho}j^{-\beta(1-\rho)}$ (now $g(x,y)= x^{-\rho\alpha}y^{-(1-\rho)\beta}$ is homogeneous, and an integral approximation argument applies provided $\alpha\rho\in (1/2,3/4)$, $\beta(1-\rho)\in (1/2,3/4)$), we should have $ \gamma\ge\rho\alpha+(1-\rho)\beta. $ By taking $\rho$ close to $1/(2\alpha)$, we expect that $\gamma\ge \beta+(\alpha-\beta)/(2\alpha)$.

Update: Matt shows below that $cn^{2-2\gamma}\le S(n)\le C n^{2-2\gamma}$, where $$\gamma=\beta+\frac{\alpha-\beta}{2\alpha}=\rho\alpha+(1-\rho)\beta\in (\alpha,\beta),$$ with $\rho=\frac{1}{2\alpha}$.